Methodology

• The study utilized an agglomerative (bottom-up) hierarchical clustering algorithm that can be described by following equation:

\[D(s_i, s_{i'}, v_i, v_{i'}) = \frac{1}{h} \sum_{j=1}^h (w_j^P \cdot d_j^P (s_{ij}, s_{i'j}) + w_j^L \cdot d_j^L (v_{ij}, v_{i'j}))\]

• \(D(s_i, s_{i'}, v_i, v_{i'})\) the global pairwise dissimilarity
• \(d_j^P(s_{ij}, s_{i'j})\) points dissimilarity for each task \(j\)
• \(d_j^L(v_{ij}, v_{i'j})\) students event patterns dissimilarity for each task \(j\)
• \(\sum_{j=1}^h w_j^P + w_j^L =1\) weight of each attribute \(h\)

• We reduce the weights for
  • R-tasks, as these tasks have more noise
  • Essay questions, as the comparison on that kind of task are limited
  • Points achieved
• Dissimilarities in points achieved for each task \(j\)

\[d_j^P(s_{ij}, s_{i'j}) = | s_{ij} - s_{i'j} |\]

• \(s_{ij}\) denotes the points achieved by student \(i\) in the \(j\)-th subtask
• Manhatten metric

• Dissimilarities in the students event patterns (time of submission) for each task \(j\)

\[d_j^L(v_{ij}, v_{i'j}) = \sum_{m=1}^{K=70} | v_{ijm} - v_{i'jm} |\]

• \(d_j^L(v_{ij}, v_{i'j})\) students event patterns dissimilarity for each task \(j\)
• Examination is divided into \(m = 1, ... , 70\) time intervals
• \(v_{ijm}\) denotes the number of answers of student \(i\) for task \(j\) in the \(m\)-th interval

Empirical Results

Figure 1: Dendogram produced by average linkage clustering of the unproctored test group (2020/21). A-F mark the clusters with the lowest dissimilarity.

• Figure 1 shows the dendrogram of the test group
  • Overall a lower level of dissimilarity compared to the comparison group
  • Six clusters (A-F) standing out noticeably from the rest of the cohort

Figure 2: Eevent logs and achieved points of the cluster B from the test group (2020/21). Above the scatter plot, a bar chart is added to compare the points per subtask.

• Figure 2 illustrates the individual comparison of achieved points and event logs of the student cluster with the highest similarity
  • Similar time path and same points for each task

Figure 3: Comparison of the normalised distance measures.

• Figure 3 compares the normalized distributions of the dissimilarity measures between the comparison and test groups

Discussion

• Three notable clusters (A, B, and E) consisting of two students each
• Collusion in larger groups are not found
• Findings the same with other linkage methods and parameter specifications as weightings
• The approach provides a basis for the examination of clusters based on comparison with a reference group